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  • Print publication year: 2010
  • Online publication date: June 2012

Three - Expeditions

Summary

The Burali-Forti paradox was a crisis for Cantor’s theory of ordinal numbers; Cantor’s paradox was a crisis for his theory of cardinals; and Russell’s paradox was a crisis for Frege’s logicism. Had the crises been local, sets (and courses-of-values) might have gone the way of phlogiston, the stuff thought in the eighteenth century to be lost from something burning (and supplanted by oxygen taken up in burning). But sets (or their kissing cousins) were not going to go without a fight. We can make out at least four reasons for this resilience. Among philosophers, logicism retained a fascination that gave it room to evolve. Among mathematicians, Cantor’s theory of infinity retained a fascination that Hilbert, for one, would not abandon. Also among mathematicians, set theory became the framework, the lingua franca, in which – by and large – modern mathematics is conducted. Finally, there are the applications of set theory, of which those in logic are central for us.

Frege layered functions. A first-level function assigns objects to objects: doubling is a first-level function that assigns six to three; and the concept:green is a first-level function that assigns truth to all and only the green things. The derivative of the square function is the doubling function, while that of the sine is the cosine, so differentiation is a second-order function. In another example, Frege reads “There are carrots,” so its subject is the concept:carrot and its predicate is the concept:existence. Existence is thus a second-level concept whose value is truth at all and only the first-level concepts under which something falls. This allows Frege to refine Kant’s criticism of the ontological argument for the existence of God. Kant said that existence is not a predicate, which is heroic, or even quixotic, grammar. Frege could say that since existence is a second-level predicate, it is at the wrong level to be a defining feature of an object like God.

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Frege, GottlobPhilosophical and Mathematical CorrespondenceOxfordBasil Blackwell 1980
Cantor, GeorgGesammelte Abhandlungen mathematischen und philosophischen InhaltsHildesheimGeorg Olms 1962
Suppes, PatrickAxiomatic Set TheoryPrincetonVan Nostrand 1960
Russell, BertrandThe Principles of MathematicsLondonGeorge Allen & Unwin 1903
Ramsey, F. P.The Foundations of Mathematics and Other Logical EssaysPaterson, N.J.Littlefield, Adams 1960
Peckhaus, VolkerOne Hundred Years of Russell’s ParadoxBerlinWalter de Gruyter 2004
Russell, BertrandLogic and Knowledge: Essays, 1901–1950LondonGeorge Allen & Unwin 1956
Wittgenstein, LudwigTractatus Logico-PhilosophicusLondonRoutledge & Kegan Paul 1961
Gödel, KurtPublications, 1938–1974OxfordOxford University Press 1990
Quine, W. V.From a Logical Point of ViewCambridge, Mass.Harvard University Press 1961
Quine, W. V.Mathematical LogicNew YorkHarper Torchbooks 1962
Whitehead, Alfred NorthRussell, BertrandPrincipia MathematicaCambridgeCambridge University Press 1935
Lipton, PeterInference to the Best ExplanationLondonRoutledge 1991
Zermelo, ErnstFrom Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931Cambridge, Mass.Harvard University Press 1967
Quine, W. V.Word and ObjectCambridge, Mass.Technology Press of the Massachusetts Institute of Technology 1960
Benacerraf, PaulPhilosophy of Mathematics: Selected ReadingsCambridgeCambridge University Press 1983
Moore, GregoryHistory and Philosophy of Modern MathematicsMinneapolisUniversity of Minnesota Press 1988
von Neumann, JohnÜber eine Widerspruchsfreiheitsfrage in der axiomatischen MengenlehreJournal für reineund angewandte Mathematik 160 1929
Barwise, JonEtchemendy, JohnThe Liar: An Essay on Truth and CircularityNew YorkOxford University Press 1987
Aczel, PeterNon-Well-Founded SetsStanfordCenter for the Study of Language and Information 1988
Gödel, KurtThe Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set TheoryPrincetonPrinceton University Press 1940
Scott, Dana S.Axiomatic Set Theory: Proceedings of Symposia in Pure MathematicsProvidenceAmerican Mathematical Society 1974
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